2 answers

Once upon a time, a little girl picked her first bunch of flowers near a happy little cottage in the country.

We usually say that someone picked flowers rather than collected them.

"Once upon a time" is assumed to be the beginning of a story. That being true, then "their" has no place in this sentence. We need to first understand who "they" are before we can refer to "their cottage."

A "little nice house" would usually be worded as a "nice little house." Since you used the words "smiley" and "cottage" in your sentence, I changed it to "happy little cottage." I don't see the need to mention that the cottage is made of wood.